Prove that <munder> <mo movablelimits="true" form="prefix">lim <mrow class="MJX-TeXAtom-O

George Bray

George Bray

Answered question

2022-06-26

Prove that lim n n n 1 n ! e n = 0

Answer & Explanation

Esteban Johnson

Esteban Johnson

Beginner2022-06-27Added 15 answers

You can use the simple fact that
e n = k = 0 n k k ! > k = n n n n !
Hence, you get
0 < n n 1 n ! e n < n n 1 n ! n n n ! = 1 n
Taniyah Estrada

Taniyah Estrada

Beginner2022-06-28Added 5 answers

By the Stirling approximation
n ! > 2 π n ( n e ) n e 1 12 n + 1
one has
0 < n n 1 n ! e n < n n 1 2 π n ( n e ) n e 1 12 n + 1 e n < 1 n .

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